This paper focus on the properties of boundary layers in periodic
homogenization in rectangular domains which are either fixed or have an
oscillating boundary. Such boundary layers are highly oscillating near the
boundary and decay exponentially fast in the interior to a non-zero limit that
we call boundary layer tail. The influence of these boundary layer tails on
interior error estimates is emphasized. Indeed, boundary layers are often more
important for improving the rate of convergence than the usual periodic
correctors. In truth we are not interested in computing exactly the full set of
boundary layers (neither are we interested in obtaining a complete asymptotic
expansion, valid at any order). Rather, we seek the non-oscillating tails of
such boundary layers away from the boundary, and we determine if their
knowledge improves, or not, the convergence rate of the homogenization process.
It turns out that these boundary layers tails can be incorporated into the
homogenized equation by adding dispersive terms and a Fourier boundary
condition. We therefore derived optimal interior estimates using these simple
boundary layer tails. Another feature of our work is that we focus on error
estimates in the $L^2$-norm rather than in the $H^1$-norm as usual. The reason
for this is that, in many applications, it is preferable to have a good
approximation of the unknown itself rather than of its gradient. Boundary
layers are often negligible for interior error estimates in the $H^1$-norm at
first order, but not in the $L^2$-norm at second-order (recall that the $L^2$
error estimate at first-order is trivial). Part of the novelty of our work
comes from this focus on higher order interior error estimates in the
$L^2$-norm.